numerical solution of flow and heat transfer over a...

Procedia Engineering 53 ( 2013 ) 542 – 554

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility of the Research Management & Innovation Centre, Universiti Malaysia Perlisdoi: 10.1016/j.proeng.2013.02.070

Malaysian Technical Universities Conference on Engineering & Technology 2012, MUCET 2012 Part 2 Mechanical And Manufacturing Engineering

Numerical Solution of Flow and Heat Transfer over a Stretching Sheet with Newtonian Heating

using the Keller Box Method N. M. Sarifª,* M. Z. Salleha and R. Nazarb

aFaculty of Industrial Science and Technology, Universiti Malaysia Pahang, 26300 UMP Kuantan, Pahang, Malaysia

bSchool of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Abstract

In this paper, the steady boundary layer flow and heat transfer over a stretching sheet with Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature is studied. The nonlinear boundary layer equations are transformed into ordinary differential equations which are then solved numerically via the Keller box method. Numerical solutions are obtained for the wall temperature and the local heat transfer coefficient for various values of the Prandtl number and the conjugate parameter . © 2013 The Authors. Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Research Management & Innovation Centre, Universiti Malaysia Perlis. Keywords: boundary layer, heat transfer, Keller box method, Newtonian heating, stretching sheet.

1. Introduction

The fluid dynamics due to the stretching sheet is important in extrusion processes. The production of sheeting material arises in a number of industrial manufacturing processes and include both metal and polymer sheets like the cooling of an infinite metallic plate in a cooling bath, paper production, glass blowing, etc. The quality of the final product depends on the rate of heat transfer at the stretching surface. The forced convection boundary layer over a stretching sheet was first studied by Crane [1]. Later, Gupta and Gupta [2] investigated heat and mass transfer on a stretching sheet with suction or blowing. In their work, the authors considered the isothermal moving plate and obtained the temperature and concentration distributions.

Merkin [3] in his work showed that wall to ambient temperature distributions can be presented by four common heating processes, namely, (i) constant or prescribed surface temperature; (ii) constant or prescribed surface heat flux; (iii) Newtonian heating; and (iv) conjugate boundary conditions. When the heat transfer from the bounding surface with a finite heat capacity is proportional to the local surface temperature, this heating process is called Newtonian heating which is usually termed conjugate convective flow. The situation of Newtonian heating occurs in many engineering devices, such as

* Corresponding author. E-mail address: [email protected]

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© 2013 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility of the Research Management & Innovation Centre, Universiti Malaysia Perlis

543 N. M. Sarif et al. / Procedia Engineering 53 ( 2013 ) 542 – 554

in heat exchanger where the solid tube wall is greatly influenced by the convection in fluid flowing over it (Chaudhary and Jain [4]). For conjugate heat transfer around fins where the conduction within the fin and the convection in the fluid surrounding it must be simultaneously analyzed in order to obtain the vital design information and also in convection flows set up by bounding surfaces absorb heat by solar radiation. Since the investigation made by Luikov et al. [5], many researches on the topic of conjugate parameter have been studied. Kimura et al. [6] and Martynenko and Khramtsov [7] have provided excellent reviews of the topic of conjugate problems in their books.

Most of the studies done by previous researchers only dealt with flow driven either by constant or prescribed surface temperature or prescribed surface heat flux, namely by Elbashbeshy [8], Hassanien et al. [9], Ishak et al. [10] and Elbashbeshy and Bazid [11]. However the Newtonian heating problem has been examined by Lesnic et al. [12] to study the free convection boundary layer along a vertical surface embedded in a porous medium. Salleh et al. [13, 14] investigated the forced convection boundary flow at a forward stagnation point with Newtonian heating as well as the boundary layer flow and heat transfer over a stretching sheet with Newtonian heating, respectively. Recently, Hayat et al. [15] addressed the effect of Newtonian heating on the boundary layer flow and heat transfer in the second grade fluid. The homotopy analysis method (HAM) has been used in their work to obtain semi-analytical solutions.

The objective of the present study is to investigate the problem of heat transfer over a stretching sheet with Newtonian heating (NH) and to see the effect of various values of Prandtl number and conjugate parameter which measures the strength of surface heating. The analysis for constant wall temperature (CWT) and constant heat flux (CHF) is also included in this paper for comparison purposes. The governing nonlinear equations are reduced into ordinary differential equations using the similarity transformation and they are then solved numerically using the Keller-box method, an implicit finite-difference scheme.

2. Governing Equation

We consider the steady laminar boundary layer flow of a viscous and incompressible fluid over a stretching sheet. The continuity, momentum and energy equations describing the flow can be written as

(1)

(2)

(3)

where x and y are taken as the coordinates parallel to the plate and normal to it, and u and v are the velocity components along the x and y directions, respectively. The associated boundary conditions are expressed as

, 0, (NH) at 0

0, as

w sT

u u ax v h T yy

u T T y (4)

where T is the fluid temperature, T is the ambient temperature, hs is the heat transfer parameter for Newtonian heating, is the kinematic viscosity and is the thermal diffusivity. The continuity equation is satisfied if we choose a stream function such that

(5)

0yv

xu

2

2

y

uyv

vxu

u

2

2

y

TyT

vxT

u

xv

yu and,


Introducing the similarity transformation

(6)

Equations (2) and (3) can be written as

(7)

Pr 0f (8)

with boundary conditions (4) become

(0) 0, (0) 1, (0) [1 (0)] (NH) at 0

( ) 0, ( ) 0 as

f f

f (9)

along with the two cases

(0) 1 (CWT) and (0) 1 (CHF) (10) where 21)/(sh is the conjugate parameter for Newtonian heating and Pr is the Prandtl number. When 0 , an insulated wall is present and when , the wall temperature remains constant.

3. The Keller Box Method

Equations (7) and (8) subject to boundary conditions (9) are solved numerically using the Keller box method as described in the books by Na [16] and Cebeci and Bradshaw [17].

3.1. The Finite Difference Scheme

This paper discusses the implicit finite-difference scheme on boundary layer flow over a stretching sheet with Newtonian heating boundary condition. We start with introducing new independent variables

),,(xu ),,(xv ),,(xt ),,(xs with , andf u u v s t , so that Equations (7) and (8) become

(11)

(12)

(NH) )(

,(CHF))(,(CWT) )(

,)(),()(21

21

21

TTT

aTTqk

TTTT

yavxfav

ww

0)( 2ffff

02ufvv

0Pr tft


Fig. 1. Net Rectangle for Difference Approximation.

We now consider the net rectangle in the x- plane shown in Figure 1 and the net points defined as below:

nnJjhJnkxxx

Jjjj

nnn

,,2,1,,0,,2,1,,0

10

10

where nk is the x spacing and jh is the spacing. Here n and j are just the sequences of numbers that indicate the coordinate location. The derivatives in the x- direction are replaced by finite difference, for example the finite difference form any points are

(a) ],[21],[

21

121

121

nj

nj

n

j

nj

nj

nj

(b)

1 1 1111 1 2 222 12 2

1 12 2

,

n nn nnn

j j j j

j n jj

u u u uu ux k h

We start writing the finite-difference form of equation for the midpoint ),( 21jnx of the segment P1P2 using centered-

difference derivatives. This process is called centering about " ),( 21jnx ".

(13a)

(13b)

(13c)

1 1 1

211 12

( )( )4

( )4

jj j j j j j

jj j j

hv v f f v v

hu u R (13d)

(13e)

0)(2 11 jj

jjj uu

hff

0)(2 11 jj

jjj vv

huu

0)(2 11 jj

jjj tt

hss

212111 ))((

4 jjjjjj

jj Rttffh

tft


where

(14a)

211

212 Pr j

j

jjjj ft

htt

hR

(14b)

We note that

211 jR and

212 jR involve only known quantities if we assume that the solution is known on 1nxx . In

terms of the new dependent variables, the boundary conditions become

( ,0) 0, ( ,0) 1, ( ,0) [1 (0)],

( , ) 0, ( , ) 0f x u x t x s

u x s x (15)

Equations (13) are imposed for j=1, 2,...,J and the transformed boundary layer thickness, j , is sufficiently large so that

it is beyond the edge of the boundary layer (Keller and Cebeci [18]). The boundary conditions yield at nxx are

0, (1 ), 0, 00 0 0 0

n n n n n nf u t u sJ J (16)

3.2.

( 1) ( ) ( ) ( 1) ( ) ( ), ,

( 1) ( ) ( ) ( 1) ( ) ( ) ,

( 1) ( ) ( )and

i i i i i if f f u u uj j j j j ji i i i i iv v v s s sj j j j j j

i i it t tj j j (17)

Substituting these expressions into Equation (13) and then drop the quadratic and higher order terms in

)()()()()( and,,,, ij

ij

ij

ij

ij tsvuf , this procedure yields the following tridiagonal system.

21311

21211

21111

2121

21

jjjjjj

jjjjjj

jjjjjj

rffhss

rvvhuu

rffhff

(18a,b,c)

214165

143121

jjjjj

jjjjjjjj

ruaua

fafavava

(18d)

212

21

1

211 jj

j

jjjj ufv

hvv

hR


(18e)

where

(19a)

(19b) and

1 1 12

2 1 12

3 1 12

214 1 1 1 12 2 2

15 2 1 1 12 2 2

,

( ) ( )

( ) Pr

j j jj j

j j jj j

j j jj j

j j jj j j j

j j jj j j j

r f f h u

r u u h v

r s s h t

r R v v h f u

r R t t h f t

(19c)

To complete the system (18) we recall the boundary conditions (16) which can be satisfied exactly with no iteration. Therefore, in order to maintain these correct values in all the iterates, we take

(20)

3.3. The Block Tridiagonal Matrix

The linearized difference equations (18) have a block tridiagonal structure consists of variables or constants, but here it consists of block matrices. In our Newtonian heating case, the elements of the matrices are defined as follows:

215143121 jjjjjjjjj rfbfbtbtb

jjjjj

jjj

jj

jjj

jj

aauha

aavh

a

aafh

a

56215

34213

12211

,

,2

2,2

1

jjj

jj

jjj

jj

bbth

b

bbfh

b

34213

12211

,Pr2

Pr2,

2Pr1

0,0,00

,00

,00 J

sJ

utuf


1 2

1 12 2 2

2 2

1 1

1 1 1

J J

J J J J J

J J

A CrB A Cr

rB A C r

B A

That is:

(21)

where

(22a)

(22b)

(23)

(24)

rA

2,

00000

001000000100

13

132

1jh

d

bbaaa

ddd

A

Jjh

d

bbaaa

dd

d

A jj 2,

2,

00000

00100001

0010

13

136

Jjh

d

bbaa

dd

B jj 2,

2,

000000

0000000000100

24

24

11,2

,

0000000000000000010000

5

Jjh

da

dd

C jj


(25)

and

(26) To solve Equation (21), we assume that A is nonsingular and it can be factored into

(27)

where

JJ

J

L

1

22

1

and II

II

U

J 1

2

1

where I is the matrix of order 5 and i and i are 55 matrices which elements are determined by the following equations:

11 A (28)

111 CA (29)

JjBA jjjj ,,3,21 (30)

Jj

tvf

su

tvftv

j

j

j

j

j

j 2,,1

1

1

1

1

0

0

1

Jj

r

r

r

r

r

r

j

j

j

j

j

j 1,

215

214

213

212

211

ULA


1,,3,2 JjC jjj (31)

Equation (27) can now be substituted into Equation (21) and we get

rUL (32) If we define

WU (33) then, Equation (32) becomes

rWL (34) where

Jj

WW

WW

W

j

J

1

1

2

1

and the ,jW are 15 column matrices. The elements W can be solved from Equation (34)

111 rW (35)

JjWBrW jjjj 212 (36)

The step in which j , j and jW are calculated is usually referred to as the forward sweep. Once the element of W are found, Equation (33) then gives the solution in the so-called backward sweep, in which the elements are obtained by the following relations:

JJ W (37)

111 JjW jjJJ (38)

These calculations are repeated until some convergence criterion is satisfied and calculations are stopped when

1)(

0iv (39)

where 1 is small prescribed value.

4. Result and Discussion

Equations (7) and (8) subject to boundary conditions (9) are solved numerically by an implicit finite-difference scheme, namely the Keller box method. Matlab software was used to programme and generate the numerical solution of the boundary value problem. The step size used in this study is 0.02 with various Prandtl number and conjugate parameters are being considered.

In order to validate the numerical results obtained, we first compare the present results with the results obtained by Hassanien et al. [9] and Elbashbeshy [8] for the case of CWT and CHF, respectively. The values of the heat transfer coefficient - (0) and surface temperature (0) from Equations (7) and (8) are presented in Table 1 and it is found that the agreement is very good.


Table 2 shows the values of (0) and (0) for various values of Pr when 1 for the case of Newtonian heating. We observe that both (0) and (0) decrease as Pr increases. The trend for NH case is similar to the CHF but different for the CWT case.

Table 1. Comparison Between The Current Solution Of Equations (7) and (8) With Previously Published Results For Different Boundary Conditions

(CWT) and (CHF)

Pr (0) (CWT) (0) (CHF)

Hassanien et al. [9]

Current Solution

Elbashbeshy [8]

Current Solution

3 1.16525 1.16525 0.85995 5 1.56805 0.63852 7 1.89540 0.52805

10 2.30801 2.30800 0.43341 0.43352 100 7.74925 7.76565 0.12852

Table 2. Values of (0) and (0) from equations (7) and (8) for various value Of PR when 1 (NH)

Pr (0) (0)

3 6.04385 7.04385

5 1.76005 2.76005

7 1.11816 2.11816

10 0.76507 1.76507

100 0.14757 1.14757

Table 3 presents the various values of with fixed 10Pr . It can be seen from this table that as increases, both )0( and (0) is also increases.

Table 4 presents the values of (0) and (0) for various values of Pr when 0.5 and 2. From this table, we noticed that as Pr increases, both (0) and (0) decrease.

Table 4. Values of (0) and (0) for various values of Pr when 0.5 and 2 (NH)

Pr 0.5 2

(0) (0) (0) (0)

3 0.75149 0.87575 - - 5 0.46815 0.73407 - - 7 0.35835 0.67917 - -

10 0.27658 0.63829 6.50263 14.971733 100 0.06875 0.53437 0.34690 2.692733

Figure 2 illustrates the variation of wall temperature (0) with Prandtl number Pr when 0.5,1 and 2.0 . To achieve an acceptable solution, Pr must be greater than some critical value, say Prc, depending on . As Pr approaches the


critical value, )0( becomes large and the value of Pr 0.8535, 2.331 and 4.851c when 0.5, 1 and 2, respectively.

Fig. 2. Variations of the (0) when 0.5, 1.0 and 1.5

Fig. 3. Variations of the wall temperature (0) when Pr 1, 3 and 7

Variations of wall temperature (0) for different values of when Pr 1, 3 and 7 are shown in Fig. 3. Also in this

case, to get an acceptable solution, must be less than some critical value, say c depending on Pr. From the graph above, the critical value of c is 0.5833 when Pr=1, 1.1547 when Pr=3, and 1.8750 when Pr=7. Fig. 4 presents the temperature profiles for various values of when Pr 10 The thermal boundary layer thickness decreases with the increasing of .

The temperature profiles with various values of Pr when 0.5 are presented in Figure 5. It is found that as Pr increases, the temperature profiles decrease. It is also shown from these figures that the thermal boundary layer thickness increases sharply with a decrease in Pr. This is because for small values of the Prandtl number, the fluid is highly conductive. Physically, if Pr increases, the thermal diffusivity decreases and this phenomenon lead to the decreasing of energy transfer ability that reduces the thermal boundary layer.

Fig. 4. Temperature profile for various values of conjugate parameter when Pr 10


Fig. 5. Temperature profiles ( ) for various values of Pr when 0.5

5. Conclusions

In this paper we have numerically studied the problem of the boundary layer flow and heat transfer over a stretching sheet with Newtonian heating. The numerical solution is obtained using the Keller box method. It is shown in this paper how the Prandtl number Pr and the conjugate paramater affect the wall temperature 0 and the heat transfer coefficient 0 . We can summarize that the thermal boundary layer thickness depends strongly on the Prandtl number and the conjugate parameter. It is found that an increase in Pr results in a decrease of the temperature profiles. However, the temperature profiles decrease by increasing conjugate parameter .

Acknowledgement

The authors gratefully acknowledge the financial support received from Universiti Malaysia Pahang (RDU110390 and

RDU110108). References [1] L. J. Crane, "Flow Past a Stretching Plate," Z. Angew. Math. Phys., vol. 21, no. 4, pp. 645-647, 1970. [2] P. S. Gupta, A. S. Gupta, "Heat and Mass Transfer on a Stretching Sheet with Suction and Blowing," Can. J. Chem. Eng., vol. 55, no. 6, pp. 744-746,

Dec. 1977. [3] J. H. Merkin, "Natural-Convection Boundary-Layer Flow on a Vertical Surface with Newtonian Heating," Int. J. Heat Fluid Flow, vol. 15, no. 5, pp.

392-398, Oct. 1994. [4] -Layer Flow Past an Impulsively Started Vertical

Surface with Newtonian Heating," J. Eng. Phys. Thermophys, vol. 80, pp. 954-960, 2007. [5] A. V. Luikov, V. A. Aleksashenko and A. A. Aleksashenko, "Analytic Methods of Solution of Conjugate Problem a in Convection Heat Transfer,"

Int. J. of Heat and Mass Transfer, vol. 14, no. 8, pp. 1047-1056, Aug. 1971. [6] S. Kimura, T. Kiwata, A. Okajima and I. Pop, "Conjugate Natural Convection in Porous Media", Adv. Water Resource, vol. 20, pp. 111-126, 1997. [7] O. G. Martynenko and P. P. Khramsov, Free Convection Heat Transfer, Springer, Berlin 2005. [8] E. M. A. Elbashbeshy, "Heat Transfer over a Stretching Surface with Variable Surface Heat Flux, J. Phys. D: Appl. Phys., vol. 31, pp. 1951-1954,

1998. [9] I. A. Hassanien, A. A. Abdullah and R. S. R. Gorla, "Flow and Heat Transfer in a Power-Law Fluid over a Nonisothermal Stretching Sheet," Math.

Comput. Model, vol. 28, no. 9, pp. 105-116, Nov. 1998. [10] A. Ishak, R. Nazar and I. Pop, "Heat Transfer over a Stretching Surface with Variable Heat Flux in Micropolar Fluids, Phys. Lett. A, vol. 372, no. 5,

pp. 559-561, Jan. 2008. [11] E. M. A. Elbashbeshy and M. A. A. Bazid, "Heat Transfer over an Unsteady Stretching Surface," Heat Mass Transfer, vol. 41, pp. 1-4, Oct. 2004. [12] D. Lesnic, D. B. Ingham and I. Pop, "Free Convection Boundary Layer Flow along a Vertical Surface in a Porous Medium with Newtonian Heating,"

Int. J. of Heat Mass Transfer, vol. 42, no. 14, pp. 2621-2627, July. 1999. [13] M. Z. Salleh, R. Nazar and I. Pop, "Forced Convection Boundary Layer Flow at a Forward Stagnation Point with Newtonian Heating," Chem. Eng.

Commun., vol. 196, no. 9, pp. 987-996, 2009


[14] M. Z. Salleh, R. Nazar, and I. Pop, "Boundary Layer Flow and Heat Transfer Over a Stretching Sheet with Newtonian Heating," J . Taiwan. Inst. Chem. Eng, vol. 41, no. 6, pp. 651-655, Nov. 2010.

[15] T. Hayat, Z. Iqbal and M. Mustafa, "Flow of a Second Grade Fluid Over a Stretching Surface with Newtonian Heating," J. Mechanics, vol. 28, no. 1, pp. 209-216, March 2012.

[16] T. Y. Na, Computational Methods in Engineering Boundary Value Problem, Academic Press, New York, 1979. [17] T. Cebeci and P. Bradshaw, Physical and Computational Aspects of Convective Heat Transfer, Springer-Verlag. New York, 1984. [18] AIAA Journal, vol. 10,

pp. 1193-1199, 1972.

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